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problem_set1 - Problem Set 1 Dynamic Meteorology/53 05 Due...

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Unformatted text preview: Problem Set 1 Dynamic Meteorology MET 43 05/53 05 .' Due: Friday 02 Sept. 2011 Basic Math 1.) Suppose we have a function f(x, y) :x2 +y27 a.) Graph this function (on one graph) for f(x,y) = 4, 9, 16. Label the isolines. b.) Derive an analytic expression (i.e., function) for the gradient in Cartesian vector component form (i.e., your equation should look something like [3 = xi + y} + 2h). c.) What is the magnitude of the gradient at f(3, 0), f(2,0),f(0,0)? (1.) Calculate the directional derivative at f(3, 0) for the meterological heading of 330°. How does this compare with the gradient at this point? e.) Calculate the gradient for the above 3 points (i.e., part c) and at f( 0, 3) using a centered finite difference, i.e. use a Taylor Series expansion to approximate the derivatives (use Ax = Ay : l). f.) Is the gradient constant or does it vary spatially? Why or why not? g.)Is the Taylor series ‘approximation’ exact for this problem (i.e. is there truncation error)? Explain. 2.) For f(x, y) : 2x2y3+ sin( 4x) - y/x2, show that 22f = 2% 6yox 6x6y 3.) Answer the following questions where: Z = Axi +Ay} +Azi}, and B = Bxi + 3)} + 32/}, and CD = q)(x, y, z) is a three-dimensional scalar quantity a.) Show that Z XE = —B ><Zl b.) Show that Z X Z = O c.) Show that (2-21)“2 = lzil (1.) Show that 2A = — is a unit vector. What direction does the unit vector point? W e.) Ifa unit vector is given by t = t1i+ t2j + t3k . For what value ofti are t1 = t2 = t3? f.) Show that V X W) = o g.) Show that v - (? xii) = O ...
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